Music Reduced to Beautiful Math

It’s hard for anyone to say what music looks like, but a new
mathematical approach sees classical music as cone-shaped and jazz as
pyramid-like.

The connections between math and music are many, from the
unproven Mozart effect (the idea that playing Mozart's music to children might improve
their mathematical abilities) to the music of the spheres (the ancient belief
that proportions in the movements of the planets could be viewed as a form of
music). Now scientists have created a mathematical system for understanding
music.

Clifton Callender of Florida State University, Ian Quinn of
Yale University and Dmitri Tymoczko of Princeton University outlined their
"geometrical music theory" in the April 18 issue of the journal Science.

The team designed a geometrical technique for mapping out music in
coordinate space. For music made of chords containing two notes, all musical
possibilities take the shape of a Möbius
strip, which basically looks like a twisted rubber band (this was first
described by Tymoczko in a 2006 Science paper). The team found that the shape
of possibilities using three-note chords is a three-dimensional ice cream cone,
where types of chords, such as major chords and minor chords, are unique points
on the cone. The space of four-note chords is what mathematicians would call a
"cone over the real projective plane," which resembles a pyramid in
our 3-D universe. Any piece of music can be mapped in these spaces.

"You can use these geometrical spaces to provide ways
of visualizing musical pieces," Tymoczko told LiveScience. "These spaces give us a much better and
comprehensive picture of the space of all possible chords."

When they first realized that the shape of two-note chords
is a Möbius strip, a fundamental
mathematical form discovered in the 19th century, the researchers
were "amazed," Quinn said.

"But there was also a sense in which we weren’t
surprised, because any composer who has spent any time futzing around on a
piano invariably finds their fingers end up twisted in a knot," he said. "Knowing
that there's a good mathematical reason for that is deeply satisfying."

It's probably no coincidence that math and music are so
deeply linked, he said.

"When music doesn't have words, it doesn't necessarily
resemble anything in the real world," Quinn said. "This is a feature
people have been amazed by and found remarkable and a little bit terrifying.
Traditionally, paintings always looked like things, poetry and literature were
talking about things. But music is coming closer to pure truth. People who talk
about mathematics say the same thing — it's not necessarily about anything,
it's just truth."

The new techniques reveal fascinating differences between
rock and classical music, and even between Paul McCartney and John Lennon.

McCartney's pieces make use of a smaller number of motions
in the geometrical spaces, corresponding to his more traditional approach to
harmony, while Lennon makes use of a much wider set of options, reflecting his
roots in rock, Tymoczko said.

"One of the really exciting things about this research
is that it allows us to see commonalities among a much wider range of musicians,"
Tymoczko said. "In some sense, Bach and
the Beatles are really exploiting the same geometrical features. In that sense
they're not radically different."

By looking at the mathematical essence behind the work of
various musicians and musical
styles, the scientists can better understand how they relate to one
another.

"You certainly see large trends," Tymoczko said.
"Over the course of the 18th and 19th centuries
people start exploring a wider variety of geometrical spaces. There's a general
push toward increasing complexity and sophistication. They move from the
three-dimensional cone to the four-dimensional space."

While analyzing the math behind music can provide many
insights, it doesn't answer all of our questions.

"A lot of people say, 'Will this help us understand
which Britney song is going to be a hit and
which one isn't?'," Tymoczko said. "There's no hope of that. There is
no way that geometry is going to help you become a great composer. Understanding
the geometry will help you become a mediocre composer much more quickly, but
composing is an artistic achievement. There's no royal road to becoming a great
musician. We're not taking the mystery away from music."

Clara Moskowitz

Clara has a bachelor's degree in astronomy and physics from Wesleyan University, and a graduate certificate in science writing from the University of California, Santa Cruz. She has written for both Space.com and Live Science.